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#1 2010-04-20 22:16:14

laipou
Member
Registered: 2009-09-19
Posts: 24

square free integer and root of unity

Which roots of unity are contained in

,where  is a square free integer?
Thanks for any help.

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#2 2010-04-21 05:51:12

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: square free integer and root of unity

d > 0 is easy.

For d < 0, we have that d^2 is in Q.  Thus, all polynomials can be represented by a + bd, where a and b are rationals.  Now you want to find all elements of this form z where z^n = 1.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2010-04-22 00:01:34

laipou
Member
Registered: 2009-09-19
Posts: 24

Re: square free integer and root of unity

Thank you for you suggestion,Ricky.
Here is the answer I think.
Let

be a primitive nth root of unity of
.
Then we know
(Euler function).
Since
,if
,then we must have
or
.
Finaly,we get
.
                   
.
                   
o,w.

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